I am planning to build two TDA7293 bridge amps. In the datasheet is written that output power of bridge is 150W (THD 10%) on 8 ohm speaker and with +/- 25V power supply.

According to the datasheet one chip can give 80W RMS (1%THD) on 4 ohm speaker with +/- 29V supply.
If in bridge every chip sees half of speaker impedance I guess I could buid +/- 29V supply for my bridge amp and get 160W on 8 ohms (1% THD) or 200W (10%THD). Than every chip would "see" 4ohms.

The main question is: Will my amp clip before with +/-29V supply, or it will just produce more more power and heat ?

You say that this chip is geting more power on 8 Om Speaker , than on a 4Om speaker?And the voltage for the second is higher..?

This is not 100% right.Normaly amps working at 8Om are giving more output power wen 4Om speakers are connected.In this case the amp is geting more hungry for input power and it heats up more.So normly in this case you have to lower your voltage to keep the output power in some normal levels or it is posible to destroy the IC..

So if you put a 8 Om speaker in bridged mode on this chip the output power will be bigger than 300W(10% THD) If you put a higher voltage the output power will grow even more.All this risks your IC.Bether lower the supply voltage and you will see no problems

Ihave recently build amp with this same chip with +/- 32V supply and it is driving 4 ohm speakers perfectly without too much heat. Now, if I put two of this chips in bridge with +/- 29V supply I think I could drive 8 ohm speaker with no problem. Every chip would act as it is working on single 4 ohm speaker and would give 100W which is normal to this chip.
Hmm, maybe I'm missing something here. Please correct me !

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So if you put a 8 Om speaker in bridged mode on this chip the output power will be bigger than 300W(10% THD)

300W??? How ? Can you please explain it better ?

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If you put a higher voltage the output power will grow even more.All this risks your IC.Bether lower the supply voltage and you will see no problems

It's not like I'm trying to connect this chips on +/- 40V or more (it can stand 120V).
Heatsinking is no problem to me.

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going back ot the start of your post.. I see a problem with the claim of 150watt into a single 8ohm load , with +-25v supply